◄UNIT 4: INDEX NUMBERS AND APPLICATIONS Topic 6►

UNIT 5: INTRODUCTION TO PROBABILITY

  • UNIT 5: INTRODUCTION TO PROBABILITY

    Key unit Competence: Use probability concepts to solve mathematical. Production, 
    financial, and economical related problems and draw 

    appropriate decisions

    Introductory activity

    The following data were collected by counting the number of Warehouses in 
    use at MAGERWA over a 20 day period. 3 of the 20 days, only 1 Warehouse 
    was used, 5 of the 20 days, 2 Warehouses were used; 8 of the 20 days, 3 
    Warehouses were used; and 4 of the 20 days, 4 warehouses were used. 
    a) Construct a probability distribution for the data 
    b) Draw a graph for the probability distribution 
    c) Show that the probability distribution satisfies the required conditions for 
    a discrete probability distribution. 

    d) On your own side defend why probability is important.

    5.1. Key Concepts of probability

     5.1.1. Definitions of probability terminologies

    Learning activity 5.1.1

    Consider the deck of 52 playing cards

    1. Suppose that you are choosing one card

    a) How many possibilities do you have for the cards to be chosen?
    b) How many possibilities do you have for the kings to be chosen?
    c) How many possibilities do you have for the aces of hearts to be 

    chosen?

    2. If “selecting a queen is an example of event, give other examples of 
    events.

    Probability is random variable that can be happen or not, i.e. is the chance 
    that something will happen. Using the following examples, we can see how the 

    concept of probability can be illustrated in various contexts

    Let us consider a game of playing cards. In a park of deck of 52 playing cards, 
    cards are divided into four suits of 13 cards each. If any player selects a card 
    by random (simple random sampling: chance of picking any card in the park is 
    always equal), then each card has the same chance or same probability of being 

    selected.

     Second example, assume that a coin is tossed, it may show Head (H-face with 

    logo) or tail(T-face with another symbol), all these are mentioned below:

    We cannot say beforehand whether it will show head up or tail up. The result 
    will depend on chance. The same, a card drawn from a well shuffled pack of 52 
    cards can be red or black. This also depends on chance. All these phenomena 
    are called probabilistic, means that can be occurred depending on the chance/

    uncertainty. The theory of probability is concerned with this type of phenomena. 

    Probability is a concept which numerically measures the degree of uncertainty 

    and therefore certainty of occurrence of events. 

    In accounting, the sample of some documents and audited, then results 
    obtained in this sample will be generalized to the whole documents with 
    supporting recommendations to that particular accountant, institutions, and 

    further decisions might be done basing on the findings obtained in the sample.

    • Random experiments and Events

    A random experiment is an experiment whose outcome cannot be predicted 
    or determined in advance. These are some example of experiments: Tossing 
    a coin, throwing a dice, selecting a card from a pack of paying cards, etc. In all 
    these cases there are a number of possible results (outcomes) which can occur 

    but there is an uncertainty as to which one of them will actually occur. 

    Each performance in a random experiment is called a trial. The result of a trial 
    in a random experiment is called an outcome, an elementary event, or a sample 
    point. The totality of all possible outcome (or sample points) of a random 
    experiment constitutes the sample space (the set of all possible outcomes of 
    a random experiment) which is denoted by Ω . Sample space may be discrete 

    (single values), or continuous (intervals). 

    Discrete sample space: 

    • Firstly, the number of possible outcomes is finite. 

    • Secondly, the number of possible outcomes is countable infinite, 
    which means that there is an infinite number of possible outcomes, 
    but the outcomes can be put in a one-to-one correspondence with the 

    positive integers.

    Continuous sample space

    If the sample space contains one or more intervals, the sample space is then 

    uncountable infinite. 

    Example: 

    A die is rolled until a “6” is obtained and the time needed to get this first “6” is 

    recorded. In this case, we have to denote that 

    An event is a subset of the sample space. The null set φ is thus an event known 
    as the impossible event. The sample space Ω corresponds to the sure event. 
    In particular, every elementary outcome is an event, and so is the sample space 

    itself.

    Note: The determination of sample space for some events such as the one for 
    dice thrown simultaneously requires the use of complex reasoning but it can 
    be facilitated by different counting techniques. Morever,the number of possible 
    arrangements for a given set is calculated mathematically, and this process is 

    known as permutation. 

    Permutation

    Permutation is a term that refers to the variety of possible arrangements or 
    orders. The arrangement’s order is important when using permutations. 
    Permutations come in three varieties, one without repetition and two with 

    repetition. There is a way you can calculate permutations using a formula

    Example 2

    Consider a portfolio manager of a bank screened out 10 companies for a new 
    fund that will consist of 3 stocks. These 3 holdings will not be equal-weighted, 
    which means that ordering will take place. Help the manager to find out the 

    number of ways to order the fund.

    Combinations

    In contrast to permutations, the combination is a method of choosing things 
    from a collection when the order of the choices is irrelevant. This means that a 
    combination is the choice of r things from a set of n things without replacement 

    and where order does not matter.

    Example 

    In a bank, the manager organizes election of bank committees consisting with 
    men and women. In how many ways a committee consisting of 5 men and 3 

    women, can be chosen from 9 men and 12 women?

    Application activity 5.1.1

    1. A box contains 5 red, 3 blue and 2 green pens. If a pen is chosen at 
    random from the box, then which of the following is an impossible 
    event? 
    a) Choosing a red pen
    b) Choosing a blue pen 
    c) Choosing a yellow pen 

    d) None of the above 

    2. Which of the following are mutually exclusive events when a day of 
    the week is chosen at random? 
    a) Choosing a Monday or choosing a Wednesday 
    b) Choosing a Saturday or choosing a Sunday 
    c) Choosing a weekday or choosing a weekend day 

    d) All of the above 

    3. Two dice are thrown simultaneously and the sum of points is noted, 

    determine the sample space. 

    5.1.2. Empirical rules, axioms and theorems

    Learning activity 5.1.2

    Consider the letters of the word “PROBABILITY”.
    a) How many letters are in this word?
    b) How many vowels are in this word? 
    c) What is the ratio of numbers of vowels to the total number of letters?
    d) How many consonants are in this word? 
    e) What is the ratio of numbers of consonants to the total number of 
    letters?

    f) Let A be the set of all vowels and B the set of all consonants. Find

    i. A ∩B 

    ii. A ∪B 

    iii. A'

    iv. B'

    Application activity 5.1.2

    1. A letter is chosen from the letters of the word “MATHEMATICS”. 

    What is the probability that the letter chosen is M? and T?

    2. An integer is chosen at random from the set 
    S = {all positive integers less than 20} . Let A be the event of choosing 

    a multiple of 3 and let B be the event of choosing an odd number. Find 

    a) P ( A∪ B)

    b) P (A ∩B )

    c) P (A − B)

    5.1.3 Additional law of probability, Mutual exclusive, and exhaustive

    Learning activity 5.1.3

    Consider a machine which manufactures car components. Suppose each 
    component falls into one of four categories: top quality, standard, 

    substandard, reject

    After many samples have been taken and tested, it is found that under certain 
    specific conditions the probability that a component falls into a category is 
    as shown in the following table. The probability of a car component falling 

    into one of four categories.

    The four categories cover all possibilities and so the probabilities must 
    sum to 1. If 100 samples are taken, then on average 18 will be top 

    quality, 65 of standard quality, 12 substandard and 5 will be rejected.

    Using the data in table determine the probability that a component selected 

    at random is either standard or top quality. 

    Solution: 

    Since only one person wins, the events are mutually exclusive.

    3. Machines A and B make components. Machine A makes 60% of the 
    Components. The probability that a component is acceptable is 0.93 when 
    made by machine A and 0.95 when made by machine B. A component is 

    picked at random. Calculate the probability that it is: 

    a) Made by machine A and is acceptable.
    b) Made by machine B and is acceptable. 

    c) Acceptable.

    Application activity 5.1.3

    1. A fair die is rolled, what is the probability of getting an even number 

    or prime number?

    2. Events A and B are such that they are both mutually exclusive and 

    exhaustive. Find the relation between these two events.

    3. In a class of a certain school, there are 12 girls and 20 boys. If a 
    teacher want to choose one student to answer the asked question
    a) What is the probability that the chosen student is a girl?
    b) What is the probability that the chosen student is a boy?
    c) If teacher doesn’t care on the gender, what is the probability of 

    choosing any student?

    4. On New Year’s Eve, the probability of a person driving while 
    intoxicated is 0.32, the probability of a person having a driving 
    accident is 0.09, and the probability of a person having a driving 

    accident while intoxicated is 0.06.

     What is the probability of a person driving while intoxicated or having a 

    driving accident?

    5.1.4 Independence, Dependence and conditional probability

    Learning activity 5.1.4

    Suppose that you have a deck of cards; then draw a card from that deck, 

    not replacing it, and then draw a second card. 

    a) What is the sample space for each event? 
    b) Suppose you select successively two cards, what is the probability of 

    selecting two red cards? 

    c) Explain if there is any relationship (Independence or dependence) 
    between those two events considering the sample space. Does the 

    selection of the first card affect the selection of the second card? 

    Contingency table

    Contingency table (or Two-Way table) provides a different way of calculating 
    probabilities. It helps in determining conditional probabilities quite easily. The 
    table displays sample values in relation to two different variables that may be 

    dependent or contingent on one another.

    Below, the contingency table shows the favorite leisure activities for 50 adults, 
    20 men and 30 women. Because entries in the table are frequency counts, the 

    table is a frequency table.


    Entries in the total row and total column are called marginal frequencies or 
    the marginal distribution. Entries in the body of the table are called joint 

    frequencies. 

    Example 

    Suppose a study of speeding violations and drivers who use car phones 

    produced the following fictional data:



    Calculate the following probabilities using the table:

    a) P(person is a car phone user)
    b) P(person had no violation in the last year)
    c) P(person had no violation in the last year AND was a car phone user)
    d) P(person is a car phone user OR person had no violation in the last year)
    e) P(person is a car phone user GIVEN person had a violation in the last 

    year)

    Application activity 5.1.4

    1. A dresser drawer contains one pair of socks with each of the 
    following colors: blue, brown, red, white and black. Each pair is 
    folded together in a matching set. You reach into the sock drawer 
    and choose a pair of socks without looking. You replace this pair 
    and then choose another pair of socks. What is the probability that 

    you will choose the red pair of socks both times?

    2. A coin is tossed and a single 6-sided die is rolled. Find the probability 

    of landing on the head side of the coin and rolling a 3 on the die. 

    4. The world-wide Insurance Company found that 53% of the residents 
    of a city had home owner’s Insurance with its company of the 
    clients, 27% also had automobile Insurance with the company. If a 
    resident is selected at random, find the probability that the resident 
    has both home owner’s and automobile Insurance with the world 

    wide Insurance Company.

    5. Calculate the probability of a 6 being rolled by a die if it is already 

    known that the result is even.

    6. A jar contains black and white marbles. Two marbles are chosen without 
    replacement. The probability of selecting a black marble and then a 
    white marble is 0.34, and the probability of selecting a black marble on 
    the first draw is 0.47. What is the probability of selecting a white marble 

    on the second draw, given that the first marble drawn was black?

    5.1.5. Tree diagram, Bayes theorem and its applications

    Learning activity 5.1.5

    1. A box contains 4 blue pens and 6 black pens. One pen is drawn at 
    random, its color is noted and the pen is replaced in the box. A pen is 

    again drawn from the box and its color is noted. 

    a) For the 1st trial, what is the probability of choosing a blue pen and 

    probability of choosing a black pen? 

    b) For the 2nd trial, what is the probability of choosing a blue pen and 
    probability of choosing a black pen? Remember that after the 1st trial 

    the pen is replaced in the box. 

    2. In the following figure complete the missing colours and probabilities

    a) Tree diagram 

    A tree diagram is a tool in the fields of probability, and statistics that helps 
    calculate the number of possible outcomes of an event or problem, and to 
    cite those potential outcomes in an organized way. It can be used to show the 
    probabilities of certain outcomes occurring when two or more trials take 
    place in succession. The outcome is written at the end of the branch and the 
    fraction on the branch gives the probability of the outcome occurring. For each
    trial the number of branches is equal to the number of possible outcomes of 

    that trial. In the diagram there are two possible outcomes, A and B, of each trial. 

    Successive trials are events which are performed one after the other; all of 

    which are mutually exclusive.

    Examples: 

    1. A bag contains 8 balls of which 3 are red and 5 are green. One ball is 
    drawn at random, its color is noted and the ball replaced in the bag. A ball 
    is again drawn from the bag and its color is noted. Find the probability 

    the ball drawn will be 

    a) Red followed by green,
    b) Red and green in any order,

    c) Of the same color

    Application activity 5.1.5

    1. Calculate the probability of three coins landing on: Three heads. 

    2. A class consists of six girls and 10 boys. If a committee of three 
    is chosen at random, find the probability of: a) Three boys being 
    chosen, b) exactly two boys and a girl being chosen. Exactly two 

    girls and a boy being chosen, d) three girls being chosen.

    3. A bag contains 7 discs, 2 of which are red and 5 are green. Two discs 
    are removed at random and their colours noted. The first disk is not 
    replaced before the second is selected. Find the probability that the 

    discs will be

    a) both red, b) of different colours, c) the same colours.

    4. Three discs are chosen at random, and without replacement, from a 
    bag containing 3 red, 8 blue and 7 white discs. Find the probability 
    that the discs chosen will be

     a) all red b) all blue c) one of each colour.

    5. 20% of a company’s employees are engineers and 20% are 
    economists. 75% of the engineers and 50% of the economists hold 
    a managerial position, while only 20% of non-engineers and non-
    economists have a similar position. What is the probability that 
    an employee selected at random will be both an engineer and a 

    manager?

    6. The probability of having an accident in a factory that triggers an 
    alarm is 0.1. The probability of its sounding after the event of an 
    incident is 0.97 and the probability of it sounding after no incident 
    has occurred is 0.02. In an event where the alarm has been triggered, 

    what is the probability that there has been no accident?

    5.2. Probability distributions

    5.2.1 Discrete probability distribution 

    Learning activity 5.2.1

    Learning activity 5.2.1

    In City of Kigali, data were collected on the number of people who had 
    applied to buy shares in the Rwanda Stock Exchange. People bought 
    different shares although some withdrew their requests. The table below 

    shows the collected data.

    Develop the probability distribution of the random variable defined as the 

    number of shares bought per person. 

    Probability Distribution: The values a random variable can assume and the 
    corresponding probabilities of each. Probability distributions are related to 

    frequency distributions.

    For a discrete random variable x, the probability distribution is defined 
    by a probability function, denoted by f(x). The probability function gives the 

    probability for each value of the random variable. Then X is a discrete random

    A discrete Variable is a variable that can take on a finite or countable number 
    of distinct values. It is a variable that is characterized by gaps in the values it 

    can take.

    Examples

    – number of heads observed in an experiment that flips a coin 10 times,
    – number of times a student visits the library;

    – number of children in class

     A discrete probability distribution consists of the values a random variable 
    can assume and the corresponding probabilities of the values. The probabilities 

    are determined theoretically or by observation.

    Two Requirements for a Probability Distribution are

    Application activity 5.2.1

    5.2.2 Binomial Distribution

    Learning activity 5.2.2

    With your potential explanations and examples to fit the following four 

    conditions:

    • A fixed number of trials 
    • Each trial is independent of the others 
    • There are only two outcomes
    • The probability of each outcome remains constant from trial to 

    trial. 

    Binomial distribution is calculated by multiplying the probability of success raised 
    to the power of the number of successes and the probability of failure raised to the 

    power of the difference between the number of successes and the number of trials.

    Let consider an experiment with independent trials and only two possible

    Application activity 5.2.2

    1. A multiple-choice test with 20 questions has five possible answers 
    for each question. A completely unprepared student picks the 
    answers for each question at random and independently. Suppose 

    X is the number of questions that the student answers correctly. 

    Calculate the probability that:

    a) The student gets every answer wrong.
    b) The student gets every answer right.

    c) The student gets 8 right answers. 

    2. An examination consisting of 10 multiple choice questions is to 
    be done by a candidate who has not revised for the exam. Each 

    question has five possible answers out of which only one is correct. 

    The candidate simply decides to guess the answers. 

    i. What is the probability that the candidate gets no answer correct?

    ii. What is the probability that the student gets two correct answers? 

    3. Suppose you took three coins and tossed them in the air. 
    a) show all the possible outcomes when the coin is tossed 
    b) What is the probability that all three will come up heads? 
    c) What is the probability of 2 heads? 
    d) Show the probability distribution on a graph (use the number of 

    heads)

    5.2.3 Bernoulli distribution

    Learning activity 5.2.3

    On your own, can you suggest the Bernoulli examples, using the binomial 

    experiment shown above? What are your differences and Similarities?

    A Bernoulli distribution is a discrete probability distribution for a Bernoulli 
    trial. A Bernoulli trial is one of the simplest experiments you can conduct. It’s 
    an experiment where you can have one of two possible outcomes. For example, 

    “Yes” and “No” or “Heads” and “Tails.

    A random experiment that has only two outcomes (usually called a “Success” 
    or a “Failure”). For example, the probability of getting a heads (a “success”) 
    while flipping a coin is 0.5. The probability of “failure” is 1 – P (1 minus the 
    probability of success, which also equals 0.5 for a coin toss). It is a special case of 
    the binomial distribution for n 1 = . In other words, it is a binomial distribution 

    with a single trial (e.g. a single coin toss).

    For example 

    In the following Bernoulli distribution, the probability of success (1) is 0.4, and 

    the probability of failure (0) is 0.6

    Application activity 5.2.3

    Provide any examples describing discrete not continuous data resulting 
    from an experiment known as Bernoulli process/trials in your areas of 

    interest as accounting professional.

    5.2.4 Poison distribution

    Learning activity 5.2.4

    Describe the characteristics of the following examples:

    a) The number of cars arriving at a service station in 1 hour; 
    b) The number of accidents in 1 day on a particular stretch of high way. 
    c) The number of visitors arriving at a party in 1 hour, 
    d) The number of repairs needed in 10 km of road,
    e) The number of words typed in 10 minutes.

    f) Why these skills and knowledge are needed in Accounting professional.

    Considering the binomial distribution, the values of p and q and n are given. 
    If there are cases where p is very small and n is very large, then calculation 
    involved will be long. Such cases will arise in connection with rare events, for 

    example.

    • Persons killed in road accidents;

    • The number of defective articles produced by a quality machine;

    • The number of mistakes committed by a good typist, per page;

    • The number of persons dying due to rare disease or snake bite;

    • The number of accidental deaths by falling from trees or roofs etc.

    In all these cases we know the number of times an event happened but not 
    how many times it does not occur. Events of these types are further illustrated 

    below:

    1. It is possible to count the number of people who died accidently by 
    falling from trees or roofs, but we do not know how many people did not 

    die by these accidents.

    2. It is possible to know or to count the number of earth quakes that 
    occurred in an area during a particular period of time, but it is, more or 
    less, impossible to tell as to how many times the earth quakes did not 

    occur.

    3. It is possible to count the number of goals scored in a foot-ball match but 

    cannot know the number of goals that could have been but not scored.

    4. It is possible to count the lightning flash by a thunderstorm but it is 
    impossible to count as to how many times, the lightning did not flash etc.
    Thus n, the total of trials in regard to a given event is not known, the binomial 

    distribution is inapplicable,

    Poisson distribution is made use of in such cases where p is very small. We 
    mean that the chance of occurrence of that event is very small. The occurrence 
    of such events is not haphazard. Their behavior can also be explained by 
    mathematical law. Poisson distribution may be obtained as a limiting case of 

    binomial distribution. 

    Application activity 5.2.4

    1. A lecturer of statistics has observed that the number of typing 
    errors in new edition of text books varies considerably from book to 
    book. After some analysis we conclude that the number of errors is 
    a Poisson distribution with a mean 1.5 per 100 pages. The lecturer 
    randomly selects 100 pages of a new book. What is the probability 

    that there are no typing errors?

    2. If there are 200 typographical errors randomly distributed in a 500-
    page manuscript, find the probability that a given page contains 

    exactly 3 errors.

    3. The mean number of bacteria per millimeter of a liquid is known 
    to be 4. Assuming that the number of bacteria follows a Poisson 
    distribution, find the probability that 1 ml of liquid there will be: No 

    bacteria, 4 bacteria, less than 3 bacteria

    5.2.5 Continuous Probability Distributions

    Learning activity 5.2.5

    A continuous random variable is a theoretical representation of a continuous 
    variable such as height, mass or time. 

    Examples
    • the amount of time to complete a task.
    • Heights of trees in a school garden
    • Daily temperature

    • Interest earned daily on bank loans per person

    Application activity 5.2.5

    5.2.6 Normal distribution

    Learning activity 5.2.6

    Explain clearly the results shown above, how these findings are linked to 
    probability distributions? Can you compare descriptive statistics and this 

    probability distribution as an accountant? How?

    Definitions

    A normal distribution is a continuous, symmetric, bell-shaped distribution of 

    a variable.

    Standard Error or the Mean: The standard deviation of the sampling 
    distribution of the sample means. It is equal to the standard deviation of the 

    population divided by the square root of the sample size. 

    Standard Normal Distribution: A normal distribution in which the mean 
    is 0 and the standard deviation is 1. It is denoted by z and Z score is used to 
    represent the standard normal distribution. 

    The formula for the standard normal distribution is

    Examples: 

    1. The age of the subscribers to a newspaper has a normal distribution with 
    mean 50 years and standard deviation 5 years. Compute the percentage 
    of subscribers who are less than 40 years old and the percentage of who 

    are between 40 and 60 years old.

    Solution:

    Step 1 Draw the figure and represent the area as shown in the following figure

    Step 3 

    Find the area, using the table which is on appendix of this student book. The 
    area under the curve to the left of z = 0.47 is 0.6808. Therefore 0.6808, or 

    68.08%, of the women spend less than $160.00 at Christmas time.

    Application activity 5.2.6

    5.3. Applications of probability and probability distributions 
    in finance, economics, businesses, and production related 

    domain.

    Learning activity 5.3

    1. The Trucks packed goods stayed at MAGERWA for under clearing 
    process in certain days. The number of days shown in the distribution are 

    displayed in the table below:

    Find these probabilities: (a) A Truck stayed exactly 5 days. (b) A Truck stayed 
    less than 6 days. (c) A Truck stayed at most 4 days. (d) A Truck stayed at least 

    5days

    Probability is essential in all aspects of human activities. Through this, we 
    can calculate families welfare, get information on different aspects of life and 
    planification in order to predict the future. It is very important to incorporate 
    probability with their properties looking at distributions to adresss these real-

    world issues. Therefore, probability play a great role for decision makers

    Examples:

    1. In Economics, A welfare of families depending on family planning. A woman 
    planning her family considers the following schemes on the assumption that 

    boys and girls are equally likely at each delivery: 

    a) Have three children.
    b) Bear children until the first girl is born or until three are born, whichever 
    is sooner, and then stop.
    c) Bear children until there is one of each sex or until there are three, 

    whichever is sooner, and then stop. 

    there are equal numbers of male and female drivers insured with the 

    Assurance Association, which selects one of them at random. 

    a) What is the probability that the selected driver makes a claim this year?

    b) What is the probability that the selected driver makes a claim in two 

    consecutive years? 

    c) If the insurance company picks a claimant at random, what is the 

    probability that this claimant makes another claim in the following year? 

    Solution: 

    Ordering such a meal requires three separate decisions:

    Choose an Appetizer: 2 choices. Choose an Entrée: 4 choices. Choose a 

    Dessert: 2 choices.

    Look at the tree diagram in the following figure. We see that, for each choice of 
    appetizer, there are 4 choices of entrees. And for each of these 24 8 × = choices, 
    there are 2 choices for dessert. A total of 2 4 2 16 ××= different meals can be 

    ordered. 

    Application activity 5.3

    1. An automatic Shaw machine fills plastic bags with a mixture of 
    beans, broccoli, and other vegetables. Most of the bags contain 
    the correct weight, but because of the slight variation in the size 
    of the beans and other vegetables, a package might be slightly 
    underweight or overweight. A check of 4000 packages filled in the 

    past month revealed:

    What is the probability that a particular package will be either underweight 

    or overweight?

    2. A baker must decide how many specialty cakes to bake each morning. 
    From past experience, she/he knows that the daily demand for 
    cakes ranges from 0 to 3.Each cake costs $3.00 to produce and 
    sells for $8.00, and any unsold cakes are thrown in the garbage at 

    the end of the day.

    a) Set up a payoff table to help the baker decide how many cakes to 

    bake

    b) Assuming probability of each event is equal, determine the expected 

    value of perfect information.

    5.4. End unit assessment

    1. Which of the following experiments does not have equally likely 
    outcomes? 
    a) Choose a number at random between 1 and 7,
    b) Toss a coin,
    c) Choose a letter at random from the word SCHOOL,

    d) None of the above.

    2. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at 
    random. What is the probability that the ticket drawn has a number 

    which is a multiple of 3 or 5? 

    3. In a class, there are 15 boys and 10 girls. Three students are 
    selected at random. What is the probability that 1 girl and 2 boys 

    are selected?

    4. One card is drawn at random from a pack of 52 cards. What is the 
    probability that the card drawn is a face card (Jack, Queen and King 

    only)?

    5. The letters of the word FACETIOUS are arranged in a row. Find the 

    probability that

    a) the first 2 letters are consonants,

    b) all the vowels are together

    6. At a certain school, the probability that a student takes Auditing 
    and Taxation is 0.087. The probability that a student takes Auditing 
    is 0.68. What is the probability that a student takes Taxation given 

    that the student is taking Auditing? 

    7. A car dealership is giving away a trip to Akagera National Park to 
    one of their 120 best customers. In this group, 65 are women, 80 
    are married and 45 married women. If the winner is married, what 

    is the probability that it is a woman?

    8. For married couples living in a certain suburb the probability that 
    the husband will vote on a bond referendum is 0.21, the probability 
    that his wife will vote in the referendum is 0.28, and the probability 
    that both the husband and wife will vote is 0.15. What is the 

    probability that

    a) at least one member of a married couple will vote ?
    b) a wife will vote, given that her husband will vote ?
    c) a husband will vote, given that his wife does not vote ?
    9. The probability that a patient recovers from a delicate heart 
    operation is 0.8. What is the probability that
    a) exactly 2 of the next 3 patients who have this operation will survive ?
    b) all of the next 3 patients who have this operation survive ?

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    ◄UNIT 4: INDEX NUMBERS AND APPLICATIONS Topic 6►